

A003231


a(n) = floor(n*(sqrt(5)+5)/2).
(Formerly M2618)


16



3, 7, 10, 14, 18, 21, 25, 28, 32, 36, 39, 43, 47, 50, 54, 57, 61, 65, 68, 72, 75, 79, 83, 86, 90, 94, 97, 101, 104, 108, 112, 115, 119, 123, 126, 130, 133, 137, 141, 144, 148, 151, 155, 159, 162, 166, 170, 173, 177, 180, 184, 188, 191, 195, 198, 202, 206, 209
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OFFSET

1,1


COMMENTS

Let r = (5  sqrt(5))/2 and s = (5 + sqrt(5))/2. Then 1/r + 1/s = 1, so that A249115 and A003231 are a pair of complementary Beatty sequences. Let tau = (1 + sqrt(5))/2, the golden ratio. Let R = {h*tau, h >= 1} and S = {k*(tau  1), k >= 1}. Then A003231(n) is the position of n*tau in the ordered union of R and S. The position of n*(tau  1) is A249115(n).  Clark Kimberling, Oct 21 2014
This is the function named c in the CarlitzScovilleVaughan link.  Eric M. Schmidt, Aug 06 2015


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=1..58.
L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337386.
Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 9.
Index entries for sequences related to Beatty sequences


FORMULA

a(n) = 2*n + A000201(n).  R. J. Mathar, Aug 22 2014


MAPLE

A003231:=n>floor(n*(sqrt(5)+5)/2): seq(A003231(n), n=1..100); # Wesley Ivan Hurt, Aug 06 2015


MATHEMATICA

With[{c=(Sqrt[5]+5)/2}, Floor[c*Range[60]]] (* Harvey P. Dale, Oct 01 2012 *)


PROG

(PARI) a(n)=floor(n*(sqrt(5)+5)/2)
(Haskell)
a003231 = floor . (/ 2) . (* (sqrt 5 + 5)) . fromIntegral
 Reinhard Zumkeller, Oct 03 2014
(MAGMA) [Floor(n*(Sqrt(5)+5)/2): n in [1..100]]; // Vincenzo Librandi, Oct 23 2014


CROSSREFS

Cf. A000201, A003231, A003233, A003234, A249115.
Sequence in context: A189999 A310188 A171983 * A189460 A172323 A190080
Adjacent sequences: A003228 A003229 A003230 * A003232 A003233 A003234


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Better description and more terms from Michael Somos, Jun 07 2000


STATUS

approved



